Scattering Equations: Real Solutions and Particles on a Line

We find $n(n-3)\mathrm{/}2$-dimensional regions of the space of kinematic invariants, where all the solutions to the scattering equations (the core of the CHY formulation of amplitudes) for $n$ massless particles are real. On these regions, the scattering equations are equivalent to the problem of finding stationary points of $n-3$ mutually repelling particles on a finite real interval with appropriate boundary conditions. This identification directly implies that for each of the $(n-3)!$ possible orderings of the $n-3$ particles on the interval, there exists one stable stationary point. Furthermore, restricting to four dimensions, we find that the separation of the solutions into $k\in \{2,3,\dots ,n-2\}$ sectors naturally matches that of permutations of $n-3$ labels into those with $k-2$ descents. This leads to a physical realization of the combinatorial meaning of the Eulerian numbers.